So, as I pointed out before, last week was the annual APS March Meeting. For those of you unfamiliar with the conference, it consists of thousands of presentations given by professors, post-doctoral students, graduate students, and even undergraduate students. Most presentations are scheduled to last ten minutes followed by two minutes of questions for the speaker. This schedule is enforced by the chair of the session. This person is tasked with deciding how many questions should be asked and informing the speaker if they are running out of time. I noticed as the week went on that the talks always seemed to be rushed toward the end and there were often very few questions allowed by the session chair. This observation brought out the scientist side of me. I decided to take data and determine how long the talks were actually taking. So, this is what I did: starting on Thursday, I timed each presentation I went to and recorded the time at the end of the presentation and then again at the end of the question period. To see the results of the experiment read on! Continue Reading

# Math

## All posts tagged Math

I was recently watching an episode of The Universe, the show from the History Channel, which detailed ten different ways to destroy the Earth (Season 4 Episode 6). One of the proposed destruction techniques was to instantly stop the rotation of the Earth. This would cause everything on the surface of the Earth to be violently thrown as the ground beneath it instantly stands still. It would be quite a violent stop. For some perspective, in New England, where I am located, you would be thrown at a velocity of about 800 miles an hour to the East. It is very unlikely that anybody could survive an impact with anything at that speed. That is not to mention that every building would be thrown at that velocity as well. However, the contributors to the show, professional astrophysicists, claimed that if you could survive the initial stop it is still unlikely that you could survive the aftermath. Specifically, the atmosphere would not be stopped, so there would be windspeeds of up to 1000 miles per hour as it continued rotating at its initial velocity. The show claims that this wind would create so much energy that it would heat up the atmosphere enough to melt rock. This claim seemed a bit too dramatic to be true, so I decided to do some calculations to see how credible it is. Brace yourselves, quite a bit of math is ahead as we fact-check…The Universe! Continue Reading

In physics, there is a famous physicist named Enrico Fermi. You may remember that I have written about him before. One of the things I mentioned in that article that is named after him is the Fermi problem. This is a type of problem that physicists love. You may have heard them referred to before as a “back of the envelope” calculation (I have also heard it referred to as “street fighting math”). That essentially sums up this kind of problem. You use facts that you know and that are rounded to nice numbers that you can easily multiply together without the use of a calculator. Often, these kinds of estimates can provide questions where a more exact answer is impossible to find. These problems are used extensively through physics (I use them all the time in research) and other sciences as well. They provide a good starting point to look for a solution to a complicated problem. To better illustrate their power, I decided to sit down and work some out showing the power of the technique. I laid out some rules for myself in this challenge:

- I cannot look up any values. I can only use numbers I know or estimate.
- I cannot use a calculator to help with the math. I must be able to do it in my head.

** Problem 1: What is the radius of the Earth?**

To start out, it is always good to use as much background knowledge as you can that you know with some certainty. So, the first fact I will utilize is that I know there are 24 time zones around the earth corresponding to the 24 hours in the day. If I can estimate the width of each of these time zones, I can find the circumference of the earth and therefore the radius. So, with that goal in mind, I know that there are four of these time zones across the continental United States. Also, it is between 3,000 and 4,000 miles across that same distance. We can estimate that at 3,500 miles, but to be easily divisible by 4, I am going to estimate it as 3,600 miles. That means that in the US, the average time zone is about 900 miles across. However, we have to remember that the US is not at the equator where they will be bigger. So, let’s add 100 miles to the width of the average time zone, that makes the average time zone at the equator a nice, easy to work with 1,000 miles. Like I said before, there are 24 time zones meaning that the circumference of the earth is approximately 24,000 miles. To get the radius from the circumference, I need to divide by 2*Pi or about 6. That means my estimate for the radius of the earth is 4,000 miles. I purposely chose the first problem to be one where there is a measured value that is easy to look up to show how accurate these estimates can be. So, dear reader, the average radius of the earth is 3,959 miles! I was only off by 41 miles! That is an error of around 1%. I am going to have to ask you to trust me that I did this honestly because I was frankly stunned I got so close after I checked.

So, now that we know this technique can provide startlingly accurate answers, let’s move on to some problems where the answer is not easy or impossible to find. For the upcoming problems, there is no “correct” answer that I could find and I had to make quite a few approximations that I am not sure how accurate they are. So, if you have a better figure for me to use or a better way to approach the problem, let me know! I am not opposed to refining estimates.

**Problem 2: How many Caucasian women in the USA make over $250,000 annually?**

To start this problem, I know there are approximately 300 million people in the US. Of those, right around half are female. That means 150 million females in the US. Then, about 60% of the population is Caucasian (I think). That means I need to take 150 million, divide by ten to get 15 million and then multiply by 6 to get 90 million. This is the approximate number of Caucasian women in the US. Then, if Barack Obama is to believed, and a tax raise on people making over $250,000 a year would only effect 2% of the population, I need to take 90 million, divide by 100 to get 0.9 million and then multiply by 2 to get 1.8 million. So, my final estimate for the number of Caucasian women making over $250,000 a year is around 1.8 million in the US. This is about 1 in every 200 people. What do you think? Does this sound about right?

**Problem 3: How many customers need to pass through a shop in Salem, MA in order for them to make a net profit?**

First, a little background on this question. My girlfriend and I recently visited Salem, MA and were stunned by the number of shops that sell the same kind of merchandise (witch memorabilia). It made me wonder how all of those shops stay in business if they are in such direct competition with each other. So, I decided to use the experience to see if they would need a huge number of customers in order to stay afloat.

Now, let’s get estimating! I am going to estimate that the average shop employs around 6 people on an average day working for 8 hours a day. This accounts for the fact that the shop is open for more than 8 hours and therefore there is not always 6 people in the store. If they are making just above minimum wage, say $9 an hour, that puts their total pay at around $500 for the day. Now, I am going to say that the store needs to sell roughly that much in merchandise each day to justify that much in wage expenses. That brings the total expenses to $1,000. Then, let’s increase that number by 50% for other expenses such as rent, utilities, and benefits bringing the total daily expenses to about $1,500. Now, I am going to estimate that the average price of an item in the store is about $20 (say for a sweatshirt). Also, it takes about ten customers going through the shop to buy one item. That means for every ten customers, the store takes in $20. So, to break even, the store needs 75 purchases of $20, or 750 customers to enter the store. This number seems a bit high to me, but when we went to the shops, it was kind of late and they were getting ready to close, so I may have not seen them very busy (it was also a day with poor weather). If anybody has retail experience at a shop like this, is this close to the mark or did I completely miss? Let me know!

**Problem 4: What is the total lost revenue for the federal government due to the current unemployment level?**

This is a pretty quick and simple one, but has a lot of ramifications related to budget debates in Washington DC. Again, I will start this problem with the estimate that there are 300 million people in the US. The current unemployment (if you haven’t been following the news) is sitting at around 10%. That means 30 million people. If the average job in the US pays around $40,000 a year (I don’t know if this is a good estimate or not), then those 30 million people making $0 would instead be making $1.2 trillion dollars if they were not unemployed. Now, I am going to guess that the average person pays a total tax rate of about 10% (again, not sure if this is good or not). That means, of that $1.2 trillion, the US government would get $120 billion. So there you have it, if everybody in the US was employed, the federal government would be about $120 billion richer every year. That number may seem large, but compared to the massive size of the budget, it is only about four percent, just a little more than a drop in the bucket.

So, now that you have seem some examples of the Fermi problem, I encourage you to try it out yourself. It is fun to try to figure out answers to crazy questions. I know Google loves to ask these kinds of problems at interviews. One I know they have asked before is “How many golf balls would fit in a school bus?” Fermi himself once came up with an estimate for the number of piano tuners in the city of Chicago. I have also found that using facts like the width of the continental US in problems like this helps you remember them better so that you can impress your friends at a trivia competition.

There is no physics in this post, but there is some math. It is easy math though. It is the kind of math that can win you a big prize on a game show. Specifically, I am thinking about the game show Let’s Make a Deal. The original host of the show was named Monty Hall, and so this math problem is usually called the Monty Hall problem.

Often in the show an audience member is chosen and they are offered three doors to choose from. The contestant wins whatever is behind the door they choose. One of the doors hides a really great prize, like a brand new car, while two of the doors have a boring, stupid goat behind them. However, the interesting bit, and what the show is really about, is that just picking a door is not the end of the game. After the contestant picks a door, good old Monty shows them a different door, one of the two the contestant did not pick, that has a goat behind it. Then, he offers the contestant a choice: stick with the door they chose or switch to the other unopened door and win what is behind it. So, what option do you think is more likely to win you a new car? Many people, in fact 87% of people according to Wikipedia, stay with the original door they chose. However, I am going to try to convince you that you double your chances of winning the car if you switch doors.

There are several different ways to try to understand this problem that are explained in the Wikipedia article, but let’s first look at a slightly different situation. Imagine, instead of three doors, there are 1000 doors. When you first pick a door, there is a 1/1000 chance you picked a car. Then, Monty opens up door after door with goats behind them, until there is only one other door left unopened. Do you switch to that door? Of course you do! Monty just eliminated 998 wrong answers for you, so the odds that a care is behind that last door are quite good, 999/1000, much better than 1/1000. The same principle applies when there are only two doors. When you make your initial guess, you have a 1/3 chance of being correct. But when Monty shows you an incorrect answer, the odds that the unopened door is the correct one is 2/3, better than the original chance you guessed right.

Perhaps an easier way to see it is this. When you make your initial guess, there is a 1/3 chance the car is behind the door you chose and there is a 2/3 chance that the car is behind one of the other two doors. When Monty opens up the door with a goat behind it, there is a zero probability there is a car there. This does not change the probability that you guessed right though. There is still just a 1/3 chance you have the right door. That means that there is still a 2/3 chance the car is not behind the door you chose, and because Monty made one of those doors a zero probability, then the remaining door has all of that 2/3 probability. So, switching doubles your chances of winning a car!

The same is true for the game show Deal Or No Deal. There are 26 cases and you get to choose one, hoping that it has the grand prize. If you get to the point where there are just two cases left, you are given a chance to swap cases. Based on the above arguments you would be a fool not to switch.

So, if anybody reading this ends up on one of these shows and does not switch when given the opportunity, just remember that I tried to inform you and give you the best chance of winning it all. Good luck!